In mathematics, the ring of integers of an algebraic number field K is the ring of all integral elements contained in K. An integral element is a root of a monic polynomial with integer coefficients, xn + cn−1xn−1 + … + c0 . This ring is often denoted by OK or \( {\mathcal O}_{K} \). Since any integer belongs to K and is an integral element of K, the ring Z is always a subring of OK.
The ring Z is the simplest possible ring of integers.[1] Namely, Z = OQ where Q is the field of rational numbers.[2] And indeed, in algebraic number theory the elements of Z are often called the "rational integers" because of this.
The ring of integers of an algebraic number field is the unique maximal order in the field.
Properties
The ring of integers OK is a finitely-generated Z-module. Indeed, it is a free Z-module, and thus has an integral basis, that is a basis b1, … ,bn ∈ OK of the Q-vector space K such that each element x in OK can be uniquely represented as
\( x=\sum _{{i=1}}^{n}a_{i}b_{i}, \)
with ai ∈ Z.[3] The rank n of OK as a free Z-module is equal to the degree of K over Q.
The rings of integers in number fields are Dedekind domains.[4]
Examples
Computational tool
A useful tool for computing the integral close of the ring of integers in an algebraic field \( K/{\mathbb {Q}} \) is using the discriminant. If K is of degree n over \( \mathbb {Q} \) , and \( {\displaystyle \alpha _{1},\ldots ,\alpha _{n}\in {\mathcal {O}}_{K}} \) form a basis of K over \( \mathbb {Q} \) , set \( {\displaystyle d=\Delta _{K/\mathbb {Q} }(\alpha _{1},\ldots ,\alpha _{n})} \) . Then, \( {\mathcal {O}}_{K} \) is a submodule of the \( \mathbb {Z} \) module spanned by \( {\displaystyle \alpha _{1}/d,\ldots ,\alpha _{n}/d} \) [5] pg. 33. In fact, if d is square-free, then this forms an integral basis for \( {\mathcal {O}}_{K} \) [5] pg. 35.
Cyclotomic extensions
If p is a prime, ζ is a pth root of unity and K = Q(ζ) is the corresponding cyclotomic field, then an integral basis of OK = Z[ζ] is given by (1, ζ, ζ2, … , ζp−2).[6]
Quadratic extensions
If d {\displaystyle d} d is a square-free integer and \( {\displaystyle K=\mathbb {Q} ({\sqrt {d}})} \) is the corresponding quadratic field, then \( {\mathcal {O}}_{K} \) is a ring of quadratic integers and its integral basis is given by (1, (1 + √d)/2) if d ≡ 1 (mod 4) and by (1, √d) if d ≡ 2, 3 (mod 4).[7] This can be found by computing the minimal polynomial of an arbitrary element \( {\displaystyle a+b{\sqrt {d}}\in \mathbb {Q} ({\sqrt {d}})} \) where \( {\displaystyle a,b\in \mathbb {Z} }. \)
Multiplicative structure
In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring of integers ℤ[√-5], the element 6 has two essentially different factorizations into irreducibles:[4][8]
\( 6=2\cdot 3=(1+{\sqrt {-5}})(1-{\sqrt {-5}})\ . \)
A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.[9]
The units of a ring of integers OK is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of K. A set of torsion-free generators is called a set of fundamental units.[10]
Generalization
One defines the ring of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤ 1; this is a ring because of the strong triangle inequality.[11] If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.[2]
For example, the p-adic integers Zp are the ring of integers of the p-adic numbers Qp .
See also
Minimal polynomial (field theory)
Integral closure - gives a technique for computing integral closures
References
Cassels, J.W.S. (1986). Local fields. London Mathematical Society Student Texts. 3. Cambridge: Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006.
Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Samuel, Pierre (1972). Algebraic number theory. Hermann/Kershaw.
Notes
The ring of integers, without specifying the field, refers to the ring Z of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "integer" in abstract algebra.
Cassels (1986) p.192
Cassels (1986) p.193
Samuel (1972) p.49
Baker. "Algebraic Number Theory" (PDF). pp. 33–35.
Samuel (1972) p.43
Samuel (1972) p.35
Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0.
Samuel (1972) p.50
Samuel (1972) pp.59-62
Cassels (1986) p. 41
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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